Algebra
🔒 Log in to trackhigh importance~3 Q in Tier 146 formulas⚡ 18 shortcuts6 subtopics
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Basic algebraic identities
Square of sum or difference
(a \pm b)^2 = a^2 \pm 2ab + b^2
Difference of squares
a^2 - b^2 = (a+b)(a-b)
Sum of cubes
a^3 + b^3 = (a+b)(a^2 - ab + b^2)
Difference of cubes
a^3 - b^3 = (a-b)(a^2 + ab + b^2)
Cube of sum
(a+b)^3 = a^3 + b^3 + 3ab(a+b)
Squares combine
(a+b)^2 + (a-b)^2 = 2(a^2+b^2)
Squares subtract
(a+b)^2 - (a-b)^2 = 4ab
Square of a trinomial
(a+b+c)^2 = a^2+b^2+c^2+2(ab+bc+ca)
Fourth powers from the ladder
a^4 + b^4 = (a^2+b^2)^2 - 2a^2b^2
$x+\frac{1}{x}$ type expressions
Square rung
x^2+\frac{1}{x^2} = \left(x+\frac1x\right)^2-2 = \left(x-\frac1x\right)^2+2
Cube rung, sum
x^3+\frac{1}{x^3} = \left(x+\frac1x\right)^3 - 3\left(x+\frac1x\right)
Cube rung, difference
x^3-\frac{1}{x^3} = \left(x-\frac1x\right)^3 + 3\left(x-\frac1x\right)
Fourth rung
x^4+\frac{1}{x^4} = \left(x^2+\frac{1}{x^2}\right)^2 - 2
Fifth rung
x^5+\frac{1}{x^5} = \left(x^2+\tfrac1{x^2}\right)\left(x^3+\tfrac1{x^3}\right)-\left(x+\tfrac1x\right)
Ladders link
\left(x+\frac1x\right)^2 - \left(x-\frac1x\right)^2 = 4
Equal-end quadratic
ax^2 - bx + a = 0 \Rightarrow x+\frac1x = \frac ba
Mixed form
\left(px+\frac{1}{qx}\right)^2 = p^2x^2+\frac{1}{q^2x^2}+\frac{2p}{q}
$a^3+b^3+c^3-3abc$ and conditional identities
Master identity
a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca)
Sum form
a^3+b^3+c^3-3abc = s\left(s^2-3P\right)
Half form
a^3+b^3+c^3-3abc = \tfrac12\,s\left[(a-b)^2+(b-c)^2+(c-a)^2\right]
Zero-sum case
a+b+c = 0 \Rightarrow a^3+b^3+c^3 = 3abc
Equal case
a^2+b^2+c^2 = ab+bc+ca \Rightarrow a=b=c
Power-sum expansion
a^3+b^3+c^3 = s^3 - 3sP + 3R
Pair products
(a+b)(b+c)(c+a) = sP - R
Pairwise from squares
ab+bc+ca = \dfrac{(a+b+c)^2-(a^2+b^2+c^2)}{2}
Surds: rationalisation and square roots of surds
Rationalisation
\frac{1}{\sqrt{a} \pm \sqrt{b}} = \frac{\sqrt{a} \mp \sqrt{b}}{a-b}
Conjugate product
(\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b}) = a-b
Root of a surd, plus
\sqrt{a+2\sqrt{b}} = \sqrt{m}+\sqrt{n},\ m+n = a,\ mn = b
Root of a surd, minus
\sqrt{a-2\sqrt{b}} = \sqrt{m}-\sqrt{n}\ \ (m > n)
Product-one pair
x = p+\sqrt{q},\ p^2-q = 1 \Rightarrow \tfrac1x = p-\sqrt{q},\ x+\tfrac1x = 2p
Telescoping sum
\sum \frac{1}{\sqrt{n}+\sqrt{n+1}} = \sqrt{\text{last}} - \sqrt{\text{first}}
Difference of roots
\sqrt{a}-\sqrt{b} = \frac{a-b}{\sqrt{a}+\sqrt{b}}
Linear equations, graphs and polynomials
Unique solution
\frac{a_1}{a_2} \ne \frac{b_1}{b_2}
lines cross once
No solution
\frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2}
parallel lines
Infinite solutions
\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}
same line
Area with the axes
\text{Area} = \frac{1}{2}\cdot\left|\frac{c}{a}\right|\cdot\left|\frac{c}{b}\right|
Remainder theorem
p(x) \div (x-a) \Rightarrow R = p(a)
for px - q, substitute q/p
Roots of a quadratic
\alpha+\beta = -\frac ba,\quad \alpha\beta = \frac ca
Discriminant
b^2 - 4ac \gtrless 0
decides root nature
Maxima and minima (AM ≥ GM, quadratics)
AM-GM
\frac{x+y}{2} \ge \sqrt{xy}
equality when x = y
Plus-form floor
ax + \frac{b}{x} \ge 2\sqrt{ab}\ \ (x>0)
at x = sqrt(b/a)
Vertex location
x = -\frac{b}{2a}
Extreme value
\frac{4ac-b^2}{4a}
max for a < 0, min for a > 0
Fixed sum
x+y = S \Rightarrow xy \le \frac{S^2}{4}
Fixed product
xy = P \Rightarrow x+y \ge 2\sqrt{P}
Always positive
ax^2+bx+c > 0 \iff a > 0,\ b^2 < 4ac
strict inequality, strict discriminant